Abstract group 11664.gv: $\He_3^2:(C_2^2\times C_4)$

• Label: 11664.gv
• Order: \( 2^{4} \cdot 3^{6} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $22$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$\He_3^2:(C_2^2\times C_4)$
Order:	\(11664\)\(\medspace = 2^{4} \cdot 3^{6} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^2.C_3^4.C_2^5.C_2^4$, of order \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \)
Composition factors:	$C_2$ x 4, $C_3$ x 6
Derived length:	$3$

This group is nonabelian and supersolvable (hence solvable and monomial).

Group statistics

Order	1	2	3	4	6	12
Elements	1	487	728	216	4616	5616	11664
Conjugacy classes	1	7	23	8	41	32	112
Divisions	1	7	22	4	40	16	90
Autjugacy classes	1	3	10	1	14	3	32

Dimension	1	2	4	8	12	18	24	36	48
Irr. complex chars.	16	32	24	12	4	16	8	0	0	112
Irr. rational chars.	8	20	24	16	4	8	4	4	2	90

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$870912$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{12}=c^{6}=d^{3}=e^{3}=f^{3}=g^{3}=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $22$ $\langle(1,2,5,10,9,6)(3,7,14,18,11,12)(4,8,15,13,17,16)(19,21,20,22), (2,4)(3,6) \!\cdots\! \rangle$
Transitive group:	36T9171				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^4$ $\,\rtimes\,$ $(C_2^2.S_3^2)$	$(\He_3^2:C_4)$ $\,\rtimes\,$ $C_2^2$	$\He_3^2$ $\,\rtimes\,$ $(C_2^2\times C_4)$	$((C_3^3\times C_6).D_6)$ $\,\rtimes\,$ $S_3$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^3:D_6)$ . $S_3^2$	$(C_3^2\times C_6)$ . $S_3^3$	$C_3^3$ . $(C_2.S_3^3)$	$C_3^2$ . $(C_6.S_3^3)$	all 16

Elements of the group are displayed as permutations of degree 22.

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 131103 subgroups in 2854 conjugacy classes, 97 normal (9 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $\He_3^2:C_2^3$
Commutator:	$G' \simeq$ $\He_3^2$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_3\times C_6$	$G/\Phi \simeq$ $C_3:S_3^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times \He_3^2$	$G/\operatorname{Fit} \simeq$ $C_2^3$
Radical:	$R \simeq$ $\He_3^2:(C_2^2\times C_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_3:S_3^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 11664: $\He_3^2:(C_2^2\times C_4)$

Order 5832: $\He_3^2:(C_2\times C_4)$ x 4, $\He_3^2:(C_2\times C_4)$ x 2, $C_2\times (C_3^2\times \He_3).D_6$

Order 3888: $C_3^4:(C_4\times D_6)$ x 4

Order 2916: $(C_3^2\times \He_3).D_6$ x 4, $\He_3^2:C_4$ x 4, $C_2\times (C_3^3.C_3^3):C_2$ x 2, $(C_3^2\times \He_3).D_6$

Order 1944: $C_3^4:(C_4\times S_3)$ x 12, $C_3^4:(C_4\times S_3)$ x 8, $C_2\times C_3^4:D_6$ x 4, $(C_3^3\times C_6).D_6$ x 4, $C_3^4:(C_2\times C_{12})$ x 4, $C_2\times C_3^3:S_3^2$ x 2

Order 1458: $\He_3^2:C_2$ x 4, $\He_3^2:C_2$ x 2, $C_2\times \He_3^2$

Order 1296: $C_6.S_3^3$ x 4, $C_6.S_3^3$ x 2

Order 972: $C_3^4:D_6$ x 16, $C_3^4:D_6$ x 13, $(C_3\times \He_3):C_{12}$ x 12, $C_3^4:C_{12}$ x 12, $C_3^3:S_3^2$ x 8, $C_2\times C_3^4:C_6$ x 4, $C_3^3:C_6^2$ x 4, $C_3^4:D_6$ x 3, $C_3^4:D_6$ x 2, $C_3^3:C_6^2$ x 2

Order 729: $\He_3^2$

Order 648: $C_3\times C_6.S_3^2$ x 14, $C_3\times C_6.S_3^2$ x 10, $C_2\times C_3^2:S_3^2$ x 9, $C_3^3:(C_4\times S_3)$ x 8, $C_3^2:(S_3\times C_{12})$ x 8, $C_3^3:(C_4\times S_3)$ x 4, $C_3^3:(C_4\times S_3)$ x 4, $C_3^4:C_2^3$ x 2, $C_2\times C_3^2:S_3^2$ x 2, $C_3^4:C_2^3$

Order 486: $C_3^4:S_3$ x 26, $C_3^4:C_6$ x 13, $C_3^4:C_6$ x 8, $C_3^4:S_3$ x 8, $C_3^4:S_3$ x 6, $C_3^4:C_6$ x 4, $C_3^4:C_6$ x 4, $C_3^4:C_6$ x 3

Order 432: $C_2.S_3^3$ x 8, $C_{12}.S_3^2$ x 4

Order 324: $C_3^3:D_6$ x 39, $C_3^2:S_3^2$ x 36, $C_3^3:C_{12}$ x 28, $C_3^3:D_6$ x 22, $C_3^3:C_{12}$ x 12, $C_3^2:C_6^2$ x 9, $C_3^2:S_3^2$ x 8, $C_3^2:S_3^2$ x 8, $C_3^3:C_{12}$ x 4, $C_3^2:S_3^2$ x 4, $C_3^2:C_6^2$ x 4, $(C_3\times \He_3):C_4$ x 4, $C_3^3:D_6$ x 3, $C_3^3:D_6$ x 2, $C_3^3:D_6$ x 2

Order 243: $C_3^4:C_3$ x 13, $C_3^2\times \He_3$ x 3

Order 216: $C_6.S_3^2$ x 20, $C_6.S_3^2$ x 16, $C_6.S_3^2$ x 14, $C_6:S_3^2$ x 12, $C_6.S_3^2$ x 10, $C_6.S_3^2$ x 10, $C_6.S_3^2$ x 9, $C_4\times C_3^2:C_6$ x 8, $C_6.S_3^2$ x 8, $C_4\times C_3^2:S_3$ x 4, $C_6:S_3^2$ x 4

Order 162: $C_3^3:S_3$ x 78, $C_6\times \He_3$ x 52, $C_3^3:C_6$ x 44, $C_3^2\wr C_2$ x 18, $C_3^3:C_6$ x 8, $C_3^3\times C_6$ x 7, $C_3^3:S_3$ x 6, $C_3^3:S_3$ x 4, $C_3^3:C_6$ x 4

Order 144: $C_4\times S_3^2$ x 8, $C_2^2.S_3^2$ x 3

Order 108: $C_3:S_3^2$ x 48, $C_3^2:D_6$ x 40, $C_3^2:D_6$ x 39, $C_3^2:D_6$ x 36, $C_3^2:C_{12}$ x 28, $C_3^2:C_{12}$ x 24, $C_3^2:C_{12}$ x 24, $C_3^2:D_6$ x 24, $C_3^2\times D_6$ x 22, $C_3:S_3^2$ x 16, $C_3^2:D_6$ x 14, $C_4\times \He_3$ x 4, $\He_3:C_4$ x 4

Order 81: $C_3\times \He_3$ x 52, $C_3^4$ x 7

Order 72: $C_6.D_6$ x 26, $S_3\times D_6$ x 21, $S_3\times C_{12}$ x 20, $C_6.D_6$ x 20, $C_{12}:S_3$ x 8, $C_6:C_{12}$ x 6, $C_6:D_6$ x 3

Order 54: $C_3^2:C_6$ x 80, $C_3^2:S_3$ x 78, $C_2\times \He_3$ x 60, $C_3^2\times C_6$ x 50, $C_3^2:C_6$ x 48, $S_3\times C_3^2$ x 44, $C_3^2:S_3$ x 28

Order 48: $C_4\times D_6$ x 6

Order 36: $S_3^2$ x 84, $C_6\times S_3$ x 80, $C_3:C_{12}$ x 52, $C_6:S_3$ x 51, $C_3\times C_{12}$ x 12, $C_3^2:C_4$ x 12, $C_6^2$ x 3

Order 27: $\He_3$ x 60, $C_3^3$ x 50

Order 24: $C_4\times S_3$ x 28, $C_2\times D_6$ x 9, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6

Order 18: $C_3\times S_3$ x 160, $C_3\times C_6$ x 96, $C_3:S_3$ x 78

Order 16: $C_2^2\times C_4$

Order 12: $D_6$ x 66, $C_{12}$ x 16, $C_3:C_4$ x 16, $C_2\times C_6$ x 9

Order 9: $C_3^2$ x 90

Order 8: $C_2\times C_4$ x 6, $C_2^3$

Order 6: $S_3$ x 60, $C_6$ x 40

Order 4: $C_2^2$ x 7, $C_4$ x 4

Order 3: $C_3$ x 22

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 11664: $\He_3^2:(C_2^2\times C_4)$

Order 5832: $C_2\times (C_3^2\times \He_3).D_6$, $\He_3^2:(C_2\times C_4)$, $\He_3^2:(C_2\times C_4)$

Order 3888: $C_3^4:(C_4\times D_6)$

Order 2916: $(C_3^2\times \He_3).D_6$, $(C_3^2\times \He_3).D_6$, $C_2\times (C_3^3.C_3^3):C_2$, $\He_3^2:C_4$

Order 1944: $C_3^4:(C_4\times S_3)$ x 2, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^4:D_6$, $(C_3^3\times C_6).D_6$, $C_3^4:(C_4\times S_3)$, $C_3^4:(C_2\times C_{12})$

Order 1458: $C_2\times \He_3^2$, $\He_3^2:C_2$, $\He_3^2:C_2$

Order 1296: $C_6.S_3^3$, $C_6.S_3^3$

Order 972: $C_3^4:D_6$ x 4, $C_3^4:D_6$ x 2, $(C_3\times \He_3):C_{12}$ x 2, $C_3^4:C_{12}$ x 2, $C_3^4:D_6$, $C_3^3:C_6^2$, $C_3^3:S_3^2$, $C_2\times C_3^4:C_6$, $C_3^3:C_6^2$, $C_3^4:D_6$

Order 729: $\He_3^2$

Order 648: $C_3\times C_6.S_3^2$ x 3, $C_2\times C_3^2:S_3^2$ x 3, $C_3\times C_6.S_3^2$ x 3, $C_3^4:C_2^3$, $C_3^4:C_2^3$, $C_3^3:(C_4\times S_3)$, $C_2\times C_3^2:S_3^2$, $C_3^3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$, $C_3^2:(S_3\times C_{12})$

Order 486: $C_3^4:C_6$ x 4, $C_3^4:S_3$ x 4, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4:S_3$

Order 432: $C_2.S_3^3$ x 2, $C_{12}.S_3^2$

Order 324: $C_3^3:D_6$ x 11, $C_3^3:D_6$ x 6, $C_3^2:C_6^2$ x 5, $C_3^3:C_{12}$ x 4, $C_3^2:S_3^2$ x 3, $C_3^3:C_{12}$ x 2, $C_3^3:D_6$ x 2, $C_3^2:C_6^2$ x 2, $C_3^3:C_{12}$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $(C_3\times \He_3):C_4$

Order 243: $C_3^4:C_3$ x 4, $C_3^2\times \He_3$ x 2

Order 216: $C_6:S_3^2$ x 4, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6:S_3^2$ x 2, $C_6.S_3^2$ x 2, $C_4\times C_3^2:S_3$, $C_4\times C_3^2:C_6$, $C_6.S_3^2$

Order 162: $C_6\times \He_3$ x 18, $C_3^3:S_3$ x 11, $C_3^3:C_6$ x 6, $C_3^2\wr C_2$ x 5, $C_3^3\times C_6$ x 4, $C_3^3:S_3$ x 2, $C_3^3:C_6$ x 2, $C_3^3:S_3$, $C_3^3:C_6$

Order 144: $C_2^2.S_3^2$ x 2, $C_4\times S_3^2$

Order 108: $C_3^2:D_6$ x 16, $C_3^2:D_6$ x 11, $C_3^2\times D_6$ x 8, $C_3^2:D_6$ x 6, $C_3^2:D_6$ x 5, $C_3^2:C_{12}$ x 4, $C_3:S_3^2$ x 4, $C_3^2:C_{12}$ x 4, $C_3^2:C_{12}$ x 4, $C_3^2:D_6$ x 3, $C_3:S_3^2$ x 2, $C_4\times \He_3$, $\He_3:C_4$

Order 81: $C_3\times \He_3$ x 18, $C_3^4$ x 4

Order 72: $S_3\times D_6$ x 6, $C_6.D_6$ x 6, $S_3\times C_{12}$ x 3, $C_6.D_6$ x 3, $C_6:D_6$ x 2, $C_6:C_{12}$ x 2, $C_{12}:S_3$

Order 54: $C_3^2\times C_6$ x 21, $C_2\times \He_3$ x 18, $C_3^2:C_6$ x 16, $C_3^2:S_3$ x 11, $S_3\times C_3^2$ x 8, $C_3^2:C_6$ x 6, $C_3^2:S_3$ x 5

Order 48: $C_4\times D_6$ x 2

Order 36: $C_6\times S_3$ x 23, $C_6:S_3$ x 16, $C_3:C_{12}$ x 9, $S_3^2$ x 6, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2, $C_6^2$ x 2

Order 27: $C_3^3$ x 21, $\He_3$ x 18

Order 24: $C_4\times S_3$ x 4, $C_2\times D_6$ x 4, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2

Order 18: $C_3\times C_6$ x 34, $C_3\times S_3$ x 23, $C_3:S_3$ x 14

Order 16: $C_2^2\times C_4$

Order 12: $D_6$ x 15, $C_2\times C_6$ x 4, $C_{12}$ x 3, $C_3:C_4$ x 3

Order 9: $C_3^2$ x 32

Order 8: $C_2\times C_4$ x 2, $C_2^3$

Order 6: $C_6$ x 14, $S_3$ x 11

Order 4: $C_2^2$ x 3, $C_4$

Order 3: $C_3$ x 10

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 11664: $\He_3^2:(C_2^2\times C_4)$ ($C_1$)

Order 5832: $\He_3^2:(C_2\times C_4)$ ($C_2$) x 4, $\He_3^2:(C_2\times C_4)$ ($C_2$) x 2, $C_2\times (C_3^2\times \He_3).D_6$ ($C_2$)

Order 2916: $(C_3^2\times \He_3).D_6$ ($C_4$) x 4, $\He_3^2:C_4$ ($C_2^2$) x 4, $C_2\times (C_3^3.C_3^3):C_2$ ($C_2^2$) x 2, $(C_3^2\times \He_3).D_6$ ($C_2^2$)

Order 1944: $(C_3^3\times C_6).D_6$ ($S_3$) x 4

Order 1458: $\He_3^2:C_2$ ($C_2\times C_4$) x 4, $\He_3^2:C_2$ ($C_2\times C_4$) x 2, $C_2\times \He_3^2$ ($C_2^3$)

Order 972: $(C_3\times \He_3):C_{12}$ ($D_6$) x 8, $C_2\times C_3^4:C_6$ ($D_6$) x 4

Order 729: $\He_3^2$ ($C_2^2\times C_4$)

Order 486: $C_3^4:C_6$ ($C_4\times S_3$) x 8, $C_3^4:C_6$ ($C_2\times D_6$) x 4

Order 324: $(C_3\times \He_3):C_4$ ($S_3^2$) x 4, $C_3^3:D_6$ ($S_3^2$) x 2

Order 243: $C_3^4:C_3$ ($C_4\times D_6$) x 4

Order 162: $C_3^3:S_3$ ($C_6.D_6$) x 4, $C_6\times \He_3$ ($S_3\times D_6$) x 4, $C_3^3\times C_6$ ($S_3\times D_6$) x 2

Order 81: $C_3\times \He_3$ ($C_4\times S_3^2$) x 4, $C_3^4$ ($C_2^2.S_3^2$) x 2

Order 54: $C_3^2\times C_6$ ($S_3^3$) x 4

Order 27: $C_3^3$ ($C_2.S_3^3$) x 4

Order 18: $C_3\times C_6$ ($C_3:S_3^3$)

Order 9: $C_3^2$ ($C_6.S_3^3$)

Order 6: $C_6$ ($C_3^2.S_3^3$) x 2

Order 3: $C_3$ ($C_3^3.(C_4\times S_3^2)$) x 2

Order 2: $C_2$ ($\He_3^2:C_2^3$)

Order 1: $C_1$ ($\He_3^2:(C_2^2\times C_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 11664: $\He_3^2:(C_2^2\times C_4)$ ($C_1$)

Order 5832: $C_2\times (C_3^2\times \He_3).D_6$ ($C_2$), $\He_3^2:(C_2\times C_4)$ ($C_2$), $\He_3^2:(C_2\times C_4)$ ($C_2$)

Order 2916: $(C_3^2\times \He_3).D_6$ ($C_4$), $(C_3^2\times \He_3).D_6$ ($C_2^2$), $C_2\times (C_3^3.C_3^3):C_2$ ($C_2^2$), $\He_3^2:C_4$ ($C_2^2$)

Order 1944: $(C_3^3\times C_6).D_6$ ($S_3$)

Order 1458: $C_2\times \He_3^2$ ($C_2^3$), $\He_3^2:C_2$ ($C_2\times C_4$), $\He_3^2:C_2$ ($C_2\times C_4$)

Order 972: $C_2\times C_3^4:C_6$ ($D_6$), $(C_3\times \He_3):C_{12}$ ($D_6$)

Order 729: $\He_3^2$ ($C_2^2\times C_4$)

Order 486: $C_3^4:C_6$ ($C_2\times D_6$), $C_3^4:C_6$ ($C_4\times S_3$)

Order 324: $C_3^3:D_6$ ($S_3^2$), $(C_3\times \He_3):C_4$ ($S_3^2$)

Order 243: $C_3^4:C_3$ ($C_4\times D_6$)

Order 162: $C_3^3\times C_6$ ($S_3\times D_6$), $C_3^3:S_3$ ($C_6.D_6$), $C_6\times \He_3$ ($S_3\times D_6$)

Order 81: $C_3^4$ ($C_2^2.S_3^2$), $C_3\times \He_3$ ($C_4\times S_3^2$)

Order 54: $C_3^2\times C_6$ ($S_3^3$)

Order 27: $C_3^3$ ($C_2.S_3^3$)

Order 18: $C_3\times C_6$ ($C_3:S_3^3$)

Order 9: $C_3^2$ ($C_6.S_3^3$)

Order 6: $C_6$ ($C_3^2.S_3^3$)

Order 3: $C_3$ ($C_3^3.(C_4\times S_3^2)$)

Order 2: $C_2$ ($\He_3^2:C_2^3$)

Order 1: $C_1$ ($\He_3^2:(C_2^2\times C_4)$)

Series

Derived series

$\He_3^2:(C_2^2\times C_4)$

$\rhd$

$\He_3^2:(C_2^2\times C_4)$

$\rhd$

$\He_3^2$

$\rhd$

$\He_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$\He_3^2:(C_2^2\times C_4)$

$\rhd$

$\He_3^2:(C_2^2\times C_4)$

$\rhd$

$C_2\times (C_3^2\times \He_3).D_6$

$\rhd$

$C_2\times (C_3^2\times \He_3).D_6$

$\rhd$

$C_2\times (C_3^3.C_3^3):C_2$

$\rhd$

$C_2\times (C_3^3.C_3^3):C_2$

$\rhd$

$C_2\times \He_3^2$

$\rhd$

$C_2\times \He_3^2$

$\rhd$

$\He_3^2$

$\rhd$

$\He_3^2$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$\He_3^2:(C_2^2\times C_4)$

$\rhd$

$\He_3^2:(C_2^2\times C_4)$

$\rhd$

$\He_3^2$

$\rhd$

$\He_3^2$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $112 \times 112$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $90 \times 90$ rational character table.